Stastny, the son of North American Ice Hockey Hall of Fame member and Slovakian forward Peter Stastny, broke the mark of 17 straight set by Winnipeg's Teemu Selanne in 1993.
The record, however, was tainted by Minnesota's 3-2 overtime win over Colorado.
"It's something special that I'll look back on probably in the summertime," Stastny said. "But right now it's tough to take after losing 3-2 in a big game for us in overtime."
His streak is the longest by any player in the league this season and the longest since Ottawa's Dany Heatley had a point in 22 straight in 2005, reports AP.
Peter Stastny had a point in 16 straight games for Quebec as a rookie in 1980-81 and Paul's uncle, Marian Stastny, had a point in 13 straight road games as a rookie with the Nordiques in 1981-82, an NHL record also broken by his nephew on Sunday.